Regularization effect on autoencoder

import time
from typing import Tuple, Dict, Union, List

import tensorflow as tf
import matplotlib.pyplot as plt
import matplotlib.cm as cm
import numpy as np
from sklearn.manifold import TSNE
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_samples, silhouette_score

NUMBER_PARAM = 10
REGULARIZER_TYPE = 'kernel_regularizer'
LS_KERNEL_REGULARIZER = np.linspace(0, 1e-4, NUMBER_PARAM)
LS_ACTIVITY_REGULARIZER = np.zeros(NUMBER_PARAM)

1. Data prep

(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()

x_train_norm = x_train/np.max(x_train)
x_test_norm = x_test/np.max(x_test)

2. Model architecture

The following model is to minimize loss function, $L_T$, in terms of loss from regular autoencoder, $L$, plus regularization term, $R$.

Encoding layer:
$h = \alpha_e(W_1 \times\ x + b_1)$, where $\alpha_e(.)$ is activation function ReLU, and the number of hidden units in $h$ is 196. Hence $h$ is a 196x1 vector in 196-dim latent space, $W_1$ is 196x748 weight matrix and $b_1$ is the bias term in the form of 196x1 vector.
Decoding layer:
$x' = \alpha_d(W_2 \times\ h + b_2)$, where $\alpha_d(.)$ is activation function sigmoid, $x'$ is the output of the autoencoder which is optimized to reconstruct back to input $x$.
Loss function:
$L_T = L + R = ||x - x'||^2 + \lambda_a\sum |h_i|^2 + \lambda_k\sum |W_1|^2$, where $i$ is the number of hidden units, $\lambda_k\ $and $\lambda_a\ $are kernel and activity regularizer respectively, l2 regularization is used.

class Autoencoder(tf.keras.Model):
    def __init__(
        self, 
        input_shape: Tuple[int,int], 
        encoding_dim: int,
        activity_regularizer: float,
        kernel_regularizer: float,
        **kwargs
    ) -> None:
        
        super().__init__(**kwargs)
        self.encoding_dim = encoding_dim
        self.encoder = tf.keras.Sequential([
            tf.keras.layers.Flatten(),
            tf.keras.layers.Dense(
                encoding_dim, 
                activation='relu', 
                activity_regularizer=tf.keras.regularizers.L2(activity_regularizer),
                kernel_regularizer=tf.keras.regularizers.L2(kernel_regularizer)
            )
        ])
        self.decoder = tf.keras.Sequential([
          tf.keras.layers.Dense(
              input_shape[0]*input_shape[1], 
              activation='sigmoid'
          ),
          tf.keras.layers.Reshape(input_shape)
        ])
        
    def call(self, inputs):
        encoded = self.encoder(inputs)
        decoded = self.decoder(encoded)
        return decoded

3. Training

def training(
    train_set: np.ndarray,
    test_set: np.ndarray,
    activity_regularizer: float,
    kernel_regularizer: float,
    encoding_dim: int,
) -> Dict[str, Union[Autoencoder, float, int, np.ndarray]]:
    
    # build
    autoencoder = Autoencoder(
        input_shape=(28, 28),
        encoding_dim=encoding_dim,
        activity_regularizer=activity_regularizer,
        kernel_regularizer=kernel_regularizer,
    )
    autoencoder.compile(optimizer='adam', loss='mse')
    callback = tf.keras.callbacks.EarlyStopping(monitor='val_loss', patience=3, min_delta=0.0001)

    # fit
    print(f'Training start:')
    print(f'activity_regularizer = {activity_regularizer}')
    print(f'kernel_regularizer = {kernel_regularizer}')
    start = time.time()
    history = autoencoder.fit(
        train_set, train_set,
        callbacks=[callback],
        epochs=100,
        batch_size=256,
        shuffle=True,
        verbose=0,
        validation_data=(test_set, test_set)
    )
    end = time.time()

    # eval
    training_loss = history.history.get('loss')[-1]
    testing_loss = history.history.get('val_loss')[-1]
    epoch = max(history.epoch)
    print('Training results:')
    print(f'training_loss = {training_loss}')
    print(f'testing_loss = {testing_loss}')
    print(f'epoch = {epoch}')
    print(f'time passed = {int(round(end-start))}s')
    print('-'*100)
    
    # return
    w1, _, _, _ = autoencoder.get_weights()
    w1_reshape = w1.T.reshape((encoding_dim,28,28))
    result = {
        'hyperparam': {
            'activity_regularizer': activity_regularizer,
            'kernel_regularizer': kernel_regularizer,
        },
        'results': {
            'training_loss': training_loss, 
            'testing_loss': testing_loss, 
            'epoch': epoch,
            'model': autoencoder,
            'w1_reshape': w1_reshape,
        }
    }
    
    return result

training_results = list()
for kernel_regularizer, activity_regularizer in zip(LS_KERNEL_REGULARIZER,LS_ACTIVITY_REGULARIZER):
    result = training(
        train_set=x_train_norm, 
        test_set=x_test_norm,
        activity_regularizer=activity_regularizer,
        kernel_regularizer=kernel_regularizer,
        encoding_dim=196
    )
    training_results.append(result)

Training start:
activity_regularizer = 0.0
kernel_regularizer = 0.0
Training results:
training_loss = 0.0013267690083011985
testing_loss = 0.001318716793321073
epoch = 28
time passed = 39s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 1.1111111111111112e-05
Training results:
training_loss = 0.0026261000894010067
testing_loss = 0.0026118617970496416
epoch = 41
time passed = 72s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 2.2222222222222223e-05
Training results:
training_loss = 0.003139795269817114
testing_loss = 0.003098108572885394
epoch = 41
time passed = 60s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 3.3333333333333335e-05
Training results:
training_loss = 0.0031682190019637346
testing_loss = 0.003135465318337083
epoch = 50
time passed = 76s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 4.4444444444444447e-05
Training results:
training_loss = 0.0035013461019843817
testing_loss = 0.003460926003754139
epoch = 48
time passed = 63s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 5.555555555555556e-05
Training results:
training_loss = 0.0037681127432733774
testing_loss = 0.00372166745364666
epoch = 47
time passed = 72s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 6.666666666666667e-05
Training results:
training_loss = 0.0039604133926332
testing_loss = 0.003910203464329243
epoch = 47
time passed = 70s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 7.777777777777778e-05
Training results:
training_loss = 0.003900613635778427
testing_loss = 0.0038728893268853426
epoch = 54
time passed = 78s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 8.888888888888889e-05
Training results:
training_loss = 0.00391967361792922
testing_loss = 0.0038747440557926893
epoch = 58
time passed = 85s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 0.0001
Training results:
training_loss = 0.004111363086849451
testing_loss = 0.004045169334858656
epoch = 56
time passed = 75s
----------------------------------------------------------------------------------------------------

4. Loss plot

MSE loss ($L_T$) against hyperparameter, kernel_regularizer ($\lambda_k$), for training set and testing set

def loss_plot(results) -> None:
    param = list()
    training_loss = list()
    testing_loss = list()
    for result in results:
        param.append(result['hyperparam'][REGULARIZER_TYPE])
        training_loss.append(result['results']['training_loss'])
        testing_loss.append(result['results']['testing_loss'])

    plt.figure(figsize=(12, 8))
    plt.plot(param, testing_loss, 'bs', label='testing_loss')
    plt.plot(param, training_loss, 'r^', label='training_loss')
    plt.ylabel('loss')
    plt.xlabel(REGULARIZER_TYPE)
    plt.title('Training loss vs testing loss')
    plt.legend(loc='upper left')
    plt.show()
    plt.close()

loss_plot(results=training_results)

5. Sparsity plot

Sparsity of $h$ against hyperparameter, kernel_regularizer ($\lambda_k$), for training set and testing set

def _sparsity_map(x: int) -> None:
    if x != 0:
        return 1
    else:
        return 0
    
_sparsity_map_vec = np.vectorize(_sparsity_map)

def plot_sparsity(
    results, 
    train_set, 
    test_set
) -> None:
    param = list()
    ls_training_sparsity = list()
    ls_testing_sparsity = list()
    for result in results:
        param.append(result['hyperparam'][REGULARIZER_TYPE])
        model = result['results']['model']
        train_encoded_imgs = model.encoder(train_set).numpy()
        test_encoded_imgs = model.encoder(test_set).numpy()
        train_sparsity = np.sum(_sparsity_map_vec(train_encoded_imgs))/train_encoded_imgs.size
        testing_sparsity = np.sum(_sparsity_map_vec(test_encoded_imgs))/test_encoded_imgs.size
        ls_training_sparsity.append(train_sparsity)
        ls_testing_sparsity.append(testing_sparsity)
        
    plt.figure(figsize=(12, 8))
    plt.plot(param, ls_testing_sparsity, 'bs', label='testing_sparsity')
    plt.plot(param, ls_training_sparsity, 'r^', label='training_sparsity')
    plt.ylabel('sparsity')
    plt.xlabel(REGULARIZER_TYPE)
    plt.title('Training sparsity vs testing sparsity')
    plt.legend(loc='upper right')
    plt.show()
    plt.close()

plot_sparsity(
    results=training_results,
    train_set=x_train_norm,
    test_set=x_test_norm,
)

6. Weight matrix of the encoder

Weight matrix of the encoder, $W_1$, is shown on a grey-scale heatmap. Each of the subplot showing a row frm $W_1$ reshaped to 28x28

def plot_w1(
    w1: np.ndarray
) -> None:
    
    w1_dim = int(np.sqrt(len(w1)))
    fig, ax = plt.subplots(
        nrows=w1_dim, 
        ncols=w1_dim, 
        figsize=(w1_dim,w1_dim)
    )
    plt.gray()

    i = 0
    for row in ax:
        for col in row:
            col.imshow(w1[i])
            col.get_xaxis().set_visible(False)
            col.get_yaxis().set_visible(False)
            i = i + 1
            
    plt.show()
    plt.close()

for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    plot_w1(
        w1=result['results']['w1_reshape']
    )

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

7. Original image vs latent space vs reconstructed image

def plot_images(
    img_gp: List[np.ndarray],
    num_img: int,
) -> None:
    
    _, ax = plt.subplots(nrows=len(img_gp), ncols=num_img, figsize=(num_img,len(img_gp)))
    plt.gray()
    for i, row in enumerate(ax):
        tmp = img_gp[i]
        for j, col in enumerate(row):
            col.imshow(tmp[j])
            col.get_xaxis().set_visible(False)
            col.get_yaxis().set_visible(False)
            
    plt.show()
    plt.close()

num_img = 20
img = x_test_norm[:num_img]
for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    
    model = result['results']['model']
    encoded_img = model.encoder(img).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    decoded_img = model.decoder(encoded_img).numpy()
    latent_dim = int(np.sqrt(len(encoded_img.T)))
    encoded_img_reshape = encoded_img.reshape((num_img,latent_dim,latent_dim))
    norm_encoded_img_reshape = norm_encoded_img.reshape((num_img,latent_dim,latent_dim))
    
    img_gp = [img, encoded_img_reshape, norm_encoded_img_reshape, decoded_img]
    plot_images(
        img_gp=img_gp,
        num_img=num_img,
    )

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

8. Latent space similarity plot

def similarity_plot(
    img: np.ndarray, 
    ls_label: List[int]
) -> None:
    
    similarity_matrix = img @ img.T
    ls_similarity = [list() for i in range(10)]
    ls_disimilarity = [list() for i in range(10)]
    for i, u in enumerate(similarity_matrix):
        for j, v in enumerate(u):
            if i == j:
                continue
            if ls_label[i] == ls_label[j]:
                label = ls_label[i]
                ls_similarity[label].append(v)
            else:
                label = ls_label[i]
                ls_disimilarity[label].append(v)
    
    plot_data = [
        {
            'data': ls_similarity,
            'title': 'Cosine similarity of the latent vector with the same label'
        },
        {
            'data': ls_disimilarity,
            'title': 'Cosine similarity of the latent vector with the different label',
        }
    ]
    _, ax = plt.subplots(nrows=1, ncols=2, figsize=(24,8))
    for idx, col in enumerate(ax):
        col.boxplot(plot_data[idx]['data'])
        col.set_title(plot_data[idx]['title'])
        col.set_ylim([0,1])
        col.set_xticks(range(1,11))
        col.set_xticklabels(range(10))
        col.set_xlabel('label')
        col.set_ylabel('cosine similarity')
        
    plt.show()
    plt.close()

num_img = 1000
img = x_test_norm[:num_img]
img_label = y_test[:num_img]
for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    model = result['results']['model']
    encoded_img = model.encoder(img).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    similarity_plot(img=norm_encoded_img, ls_label=img_label)

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

9. Latent space in T-SNE space

def tsne_plot(
    space_gp: List[np.ndarray],
    label: List[int]
) -> None:
    
    _, ax = plt.subplots(nrows=1, ncols=2, figsize=(20,10))
    for idx, col in enumerate(ax):
        space = space_gp[idx]['space']
        scatter = col.scatter(space[:,0], space[:,1], c=label, cmap='Spectral')
        col.set_title(space_gp[idx]['title'])
        col.set_xlabel('tsne 1')
        col.set_ylabel('tsne 2')
        col.legend(*scatter.legend_elements())
    plt.show()
    plt.close()

for result in training_results:
    model = result['results']['model']
    hyperparam = result['hyperparam']
    print(hyperparam)
    encoded_img = model.encoder(x_test_norm).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    print('fitting tsne for encoded_img')
    tsne_space = TSNE(n_components=2, n_jobs=-1).fit_transform(encoded_img)
    print('fitting tsne for norm_encoded_img')
    tsne_space_norm = TSNE(n_components=2, n_jobs=-1).fit_transform(norm_encoded_img)
    space_gp = [
        {
            'space': tsne_space,
            'title': 'Latest space',
        },
        {
            'space': tsne_space_norm,
            'title': 'Normalized latest space',
        },
    ]
    tsne_plot(space_gp=space_gp, label=y_test)

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

10. K-Mean plot

def kmean_plot(
    img: np.ndarray,
    max_n_cluster: int = 10,
) -> None:
    
    _, ax = plt.subplots(nrows=2, ncols=5, figsize=(25,12))
    ax = ax.flatten()
    range_clusters = range(2,max_n_cluster+1)
    for n_clusters in range_clusters:
        kmeans = KMeans(n_clusters=n_clusters)
        kmeans.fit(img)
        labels = kmeans.labels_
        avg_silhouette_score = silhouette_score(img, labels, metric='euclidean')
        sample_silhouette_values = silhouette_samples(img, labels)
        y_lower = 10
        for i in range(n_clusters):
            ith_cluster_silhouette_values = sample_silhouette_values[labels == i]
            ith_cluster_silhouette_values.sort()
            size_cluster_i = ith_cluster_silhouette_values.shape[0]
            y_upper = y_lower + size_cluster_i
            color = cm.nipy_spectral(float(i) / n_clusters)
            ax[n_clusters-1].fill_betweenx(
                np.arange(y_lower, y_upper),
                0, ith_cluster_silhouette_values,
                facecolor=color, 
                edgecolor=color, 
                alpha=0.7
            )
            ax[n_clusters-1].set_xlim([-0.1, 1])
            ax[n_clusters-1].set_ylim([0, len(img) + (n_clusters + 1) * 10])
            ax[n_clusters-1].text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))
            ax[n_clusters-1].set_xlabel('silhouette score')
            ax[n_clusters-1].set_ylabel('cluster label')
            ax[n_clusters-1].set_title(f'{n_clusters}-clusters')
            ax[n_clusters-1].axvline(x=avg_silhouette_score, color="red", linestyle="--")
            ax[n_clusters-1].set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
            ax[n_clusters-1].set_yticks([])
            y_lower = y_upper + 10

    plt.show()
    plt.close()

ls_inertia = list()
for result in training_results:
    model = result['results']['model']
    hyperparam = result['hyperparam']
    print(hyperparam)
    encoded_img = model.encoder(x_test_norm).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    space_gp = [
        {
            'space': encoded_img,
            'title': 'Latest space',
        },
        {
            'space': norm_encoded_img,
            'title': 'Normalized latest space',
        },
    ]
    for space in space_gp:
        print(f'img = {space["title"]}')
        kmean_plot(img=space['space'])

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}
img = Latest space

img = Normalized latest space